Applicable Analysis and Discrete Mathematics 2015 Volume 9, Issue 2, Pages: 245-270
https://doi.org/10.2298/AADM150930019H
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Dynamics of a nonlinear discrete population model with jumps
Higgins R.J. (Texas Tech University, Department of Mathematics and Statistics, Lubbock, USA)
Kent C.M. (Virginia Commonwealth University, Department of Mathematics, Richmond, USA)
Kocic V.L. (Xavier University of Louisiana, Department of Mathematics, New Orleans, USA)
Kostrov Y. (Xavier University of Louisiana, Department of Mathematics, New Orleans, USA)
Our aim is to investigate the global asymptotic behavior, the existence of
invariant intervals, oscillatory behavior, structure of semicycles, and
periodicity of a nonlinear discrete population model of the form xn+1= F(xn);
for n = 0,1,...,where x0> 0; and the function F is a positive piecewise
continuous function with two jump discontinuities satisfying some additional
conditions. The motivation for study of this general model was inspired by
the classical Williamson's discontinuous population model, some recent
results about the dynamics of the discontinuous Beverton-Holt model, and
applications of discontinuous maps to the West Nile epidemic model. In the
first section we introduce the population model which is a focal point of
this paper. We provide background information including a summary of related
results, a comparison between characteristics of continuous and discontinuous
population models (with and without the Allee-type effect), and a
justification of hypotheses introduced in the model. In addition we review
some basic concepts and formulate known results which will be used later in
the paper. The second and third sections are dedicated to the study of the
dynamics and the qualitative analysis of solutions of the model in two
distinct cases. An example, illustrating the obtained results, together with
some computer experiments that provide deeper insight into the dynamics of
the model are presented in the fourth section. Finally, in the last section
we formulate three open problems and provide some concluding remarks.
Keywords: oscillation and semicycles, periodicity, invariant interval, discontinuous population model, bifurcations